We study the problem of noisy sparse array interpolation, where a large virtual array is synthetically generated by interpolating missing sensors using matrix completion techniques that promote low rank. The current understanding is quite limited regarding the effect of the (sparse) array geometry on the angle estimation error (post interpolation) of these methods. In this paper, we make advances towards solidifying this understanding by revealing the role of the physical beampattern of the sparse array on the performance of low rank matrix completion techniques. When the beampattern is analytically tractable (such as for uniform linear arrays and nested arrays), our analysis provides concrete and interpretable bounds on the scaling of the angular error as a function of the number of sensors, and demonstrates the effectiveness of nested arrays in presence of noise and a single temporal snapshot.

We study single-snapshot nested array interpolation with positive sources. The problem of sparse array interpolation is traditionally cast as a low-rank Toeplitz/Hankel matrix completion problem from partial observations. In recent work, we provided the first necessary and sufficient guarantees for nested array interpolation with real measurements in the rank minimization framework. In this work, we strengthen the sufficiency results by proving that in case of positive sources it is possible to interpolate the nested array by performing a simple convex feasibility search instead of solving a rank minimization problem. Simulations demonstrate that this framework is also effective for noisy measurements, and that noisy nested array interpolation outperforms ULA extrapolation.1

This paper proposes a novel method to focus or align array manifolds at different frequencies to a single reference frequency in wideband direction of arrival (DOA) estimation. Unlike existing methods, our focusing can be performed without explicitly constructing focusing matrices, or requiring any preliminary DOA estimates. Instead the focusing is done implicitly by obtaining focused measurements as the solution to a rank minimization procedure. This paper also provides theoretical guarantees for exact focusing via rank minimization. We call this procedure Initialization-Free Implicit-Focusing (IF2). Numerical simulations are provided to demonstrate the resilience of IF2in various SNR regimes compared to past and recent wideband DOA recovery methods, and its lack of error saturation in high SNR regimes1.

This paper considers the problem of recovering the joint support (of size K) of a set of unknown sparse vectors in ℝd, each of which can be sensed using a different measurement matrix. Such models have wide applicability ranging from communication to multi-task learning. We develop an adaptive strategy called Adaptive Joint Support Recovery (Ada- JSR) that enables exact support recovery in the extreme compression regime with only m = 1 measurement per unknown vector while requiring a total complexity of no more than K⌈log2(d)⌉ measurements. Unlike existing support recovery techniques which require suitable assumptions on the correlation structure or distribution of the unknown signals in order to operate in the regime m1

We study the problem of completing the virtual array of a nested array with a single snapshot. This involves synthesizing a virtual uniform linear array (ULA) with the same aperture as the nested array by estimating (or interpolating) the missing measurements. A popular approach for virtual array synthesis involves completing a certain Hankel/Toeplitz matrix from partial observations, by seeking low-rank solutions. However, existing theoretical guarantees for such structured rank minimization (which mostly provide sufficient conditions) do not readily extend to nested arrays. We provide the first necessary and sufficient conditions under which it is possible to exactly complete the virtual array of a nested array by minimizing the rank of a certain Toeplitz matrix constructed using a single temporal snapshot. Our results exploit the geometry of nested arrays and do not depend on the source configuration or on the separation between sources.

This paper investigates the problem of recovering a binary-valued signal from compressed measurements of its convolution with a known finite impulse response filter. We show that it is possible to attain optimum sample complexity for exact recovery (in absence of noise) with a computationally efficient algorithm. We achieve this by adopting an algorithm-measurement co-design strategy where the measurement matrix is designed as a function of the filter, such that the recovery of binary signals with arbitrary sparsity is possible by using a sequential decoding algorithm. Such a filter-dependent sampler design can overcome the computational challenges associated with enforcing binary constraints, and enable us to operate in “extreme compression” regimes, where the number of measurements can be much smaller than the sparsity level.

This paper considers the problem of recovering a unipolar finite-valued signal from compressive measurements of its convolution with a known finite impulse response filter. We show that owing to the finite-value constraint the problem remains identifiable if the downsampling factor is smaller than the filter length. We develop a new computationally efficient decoding algorithm that can operate at the optimal downsampling factor under mild conditions on the filter. This allows us to explicitly impose the finite value constraint (no relaxation) without compromising on the computational tractability.1

We propose a new framework for leveraging the degrees of freedom in the difference set of a nested array with limited snapshots. Typically, this difference set (or coarray) is realized by first estimating a virtual coarray covariance matrix from the sample covariance matrix. However, with only a few snapshots, these techniques incur large estimation error, which saturates away from zero even as the signal-to-noise ratio (SNR) tends to infinity. We address this issue by moving away from estimating the coarray covariance matrix when snapshots are limited, and instead proposing a "proxy covariance matrix" (Prox-Cov) that provides an alternate estimate of the coarray subspace (but does not attempt to estimate the source powers). (Prox-Cov) is shown to outperform coarray MUSIC with limited snapshots when the number of sources exceeds the number of sensors. Moreover, when the number of sources is fewer than sensors, we prove that (Prox-Cov) leads to exact identification of the coarray subspace with very few snapshots in the absence of noise, while the error of coarray MUSIC provably saturates in this regime.

This paper considers the problem of blind deconvolution where the input signal is non-negative and sparse, and the unknown convolutional kernel is a first order autoregressive filter. Our objective is to understand if it is possible to recover both the signal and the kernel from downsampled measurements of their convolution. This work is motivated by the problem of neural spike deconvolution from calcium imaging, where it is desirable to recover spikes at a higher rate from uniformly undersampled measurements. Assuming that the signals are generated according to a Bernoulli model, we show that it is possible to uniquely identify both the signal and the kernel with high probability using only O s measurements, where s is the expected sparsity. The key p qidea is to exploit non-negative constraints on the input signal as well as the parametric structure of the kernel1.